moment based formulation of Navier-Maxwell slip boundary condition

Lattice Boltzmann Method LBM: Theory
#1

Hello everyone,

I am reading a recent paper by T. Reis and P. J. dellar in Physics of Fluids(http://pof.aip.org/resource/1/phfle6/v24/i11/p112001_s1). They have given the expression for the unknowns at the south boundary f2, f5 and f6 in equation 30(a), 30(b) and 30© respectively. Following the mathematical method explained by them i got the same expression for f2. But i think the authors have missed “-rho * uslip /2” term in equation 30(b) and “rho *uslip/2” term in equation 30©. It is very possible that i am completely wrong. But i think without these terms it will not be possible to satisfy tangential velocity
boundary condition.

Thanks

#2

Yes, you are right. There should be +/-rho*uslip/2 terms there. You spotted a typo that the authors missed in the proof-reading!!

#3

Thanks a lot Pleb

I am also getting a miss-match in the slip velocity expression at the south boundary given in eqn 31. I am getting the bracketed term in the denominator as (1 + 2tau - 6sigma* l) .

The unknowns at the south-west corner are given in eqn 32(a) to 32(e). The expressions for f1, f2 and f8 are fine. I am getting the first term of RHS of eqn 32© as " - 2 *rho_in /3" and the bracketed part of last term as “u_slip - 1”. Also, in the expression for f6 (eqn 32d), the bracketed part of last term as “u_slip + 1”.

I will be glad if you can confirm it, waiting for your response.

#4

32© should be -2rho/3, but I think the rest is ok. In your fist message I think f5 (30b) should have the +rhou/2 term, and f6 the negative.
For eqn 31, rhou=-3sigmaLPi/tau.
Since Pi=2taubar{Pi}/(2tau+1)
and bar{Pi}=rhou-f1+f3+2f7-2f8 (where all f are fbar)
you get
rhou(2tau+1)=-6sigmaL(rho*u-…)
where the … are the f terms. I think this rearranges into eqn 31.

#5

Thanks a lot sir. Sorry sir, I did not know that you are the first author of the paper.

The problem in the “sign” of the terms was becuase of a horrible mistake( in the sign of components of microscopic velocity vector), which I was previously doing. Now i am getting eqn 31 as the expression for slip velocity. As you have mentioned everything is OK apart from
“rho * u_slip/2” in eqn 30 (b), “-rho * u_slip/2” in eqn 30 © and “-2 * rho/3” in eqn 32©.

Sorry for the trouble sir and thanks a lot for the clarification.

#6

No problem at all - I apologise for the typo!
Good luck,
Tim